Time-reversible and norm-conserving high-order integrators for the nonlinear time-dependent Schr\"{o}dinger equation: Application to local control theory

TL;DR

This paper introduces high-order, time-reversible, and norm-conserving geometric integrators for nonlinear time-dependent Schr"{o}dinger equations, improving accuracy and efficiency over existing methods, especially in local control applications.

Contribution

The authors develop and analytically prove high-order geometric integrators based on symmetric compositions of the implicit midpoint method, suitable for general nonlinear Schr"{o}dinger equations with improved efficiency.

Findings

Higher-order integrators outperform second-order methods in accuracy.

Eighth-order algorithm achieves 48-fold speedup over second-order methods.

Significant computational savings in local control of a retinal model.

Abstract

The explicit split-operator algorithm has been extensively used for solving not only linear but also nonlinear time-dependent Schr\"{o}dinger equations. When applied to the nonlinear Gross-Pitaevskii equation, the method remains time-reversible, norm-conserving, and retains its second-order accuracy in the time step. However, this algorithm is not suitable for all types of nonlinear Schr\"{o}dinger equations. Indeed, we demonstrate that local control theory, a technique for the quantum control of a molecular state, translates into a nonlinear Schr\"{o}dinger equation with a more general nonlinearity, for which the explicit split-operator algorithm loses time reversibility and efficiency (because it has only first-order accuracy). Similarly, the trapezoidal rule (the Crank-Nicolson method), while time-reversible, does not conserve the norm of the state propagated by a nonlinear Schr\"{o}dinger equation. To overcome these issues, we present high-order geometric integrators suitable for general time-dependent nonlinear Schr\"{o}dinger equations and also applicable to nonseparable Hamiltonians. These integrators, based on the symmetric compositions of the implicit midpoint method, are both norm-conserving and time-reversible. The geometric properties of the integrators are proven analytically and demonstrated numerically on the local control of a two-dimensional model of retinal. For highly accurate calculations, the higher-order integrators are more efficient. For example, for a wavefunction error of $10^{-9}$, using the eighth-order algorithm yields a $48$-fold speedup over the second-order implicit midpoint method and trapezoidal rule, and $400000$-fold speedup over the explicit split-operator algorithm.